Part 7

The steel for the manufacture of steel dies is carefully selected, forged at a high heat into the rough die, softened by careful annealing, and then handed over to the engraver. After the engraver has worked out the design in intaglio, the die is put through the operation of hardening, after which, being cleaned and polished, it is called a _matrix_. This is not, however, generally employed in multiplying impressions, but is used for making a _punch_ or steel impression for relief. For this purpose another block of steel of the same quality is selected, and, being carefully annealed or softened, is compressed by proper machinery upon the matrix till it receives the impression. When this process is complete, the impression is retouched by the engraver, and hardened and collared like the matrix. Any number of dies may now be made from this punch by impressing upon it plugs of soft steel.

There is hardly any article which does not in the course of its manufacture require the use of a die of some kind. For all sorts of metal-work, seals, rings, silverware, moulds and shapes of sheet steel or tin, dies are employed. For this class of work they are usually of steel. For embossing articles of leather, wood, celluloid, rubber, cloth, or clay, dies of brass and phosphor-bronze are commonly used, these being easier media to work in and yet sufficiently strong. The dies for letter headings and company seals are cut, in reverse to the design required, in steel; those for sealing-wax seals in steel or brass; the lettering being usually punched in by hand by separate letter punches, which themselves have been cut in relief on steel. Designs which are unusually ornate may be engraved by hand. Dies for embossing designs on leather, catalogue covers, cardboard articles, cards, and soft materials are usually modelled in brass. The design in reverse is cut out to a depth corresponding to the relief wanted. These dies are usually worked up by hand by the engraver. Dies for the reproduction of rubber stamps for printing on clay are cut in phosphor-bronze or hard brass in relief and reverse, and with an extreme bevel. The dies or blocks are then struck deeply into lead, and melted rubber is poured into the moulds so formed. When set, the rubber is removed and mounted as a hand-stamp ready to impress the clay in ink. Dies for wallpapers are cut on rollers. Steel dies for flower-shapes have a cutting edge, so that they can stamp out and emboss in one action. Of late years machinery has come much into use for relieving the engraver of some of his labour, but the designs are generally kept secret. One machine, called the pantograph or engraving machine, reproduces engravings in all metals and many shapes from patterns. Many of the stamp-duty steel dies made and issued by the Royal Mint are reproduced from this machine in reductions from brass patterns.--BIBLIOGRAPHY: Lucas, _Dies and Die-making_; J. V. Woodworth, _Dies: their Construction and Use_.

DIESEL, Rudolph, German inventor, born in Paris in 1858, died in 1913. Educated in England and at Munich, he proposed in 1893 to utilize directly the energy created by the combustion of fuel, a proposal which led to his invention of the Diesel engine. (See _Internal Combustion Engines_.) In 1913 he was called to England to consult with the Admiralty on the application of his motor, but was drowned in crossing the Channel. In 1894 he published a monograph entitled _Theory and Construction of a Rational Heat Motor_.

DIES FASTI ET NEFASTI, a Roman division of days, with reference to judicial business, into working-days and holidays. A _dies fastus_ was a day on which courts and assemblies could be held and judgments pronounced; a _dies nefastus_, a day on which courts could not be held nor judgments pronounced.

DIES IRAE (d[=i]'es [=i]'r[=e]), one of the great Latin hymns of the Mediaeval Church, generally used as part of the requiem or mass for the souls of the dead. It describes, as its name ('the day of wrath') denotes, the Last Judgment of the world, and seems to have been suggested by the description in _Zephaniah_, i, 15 and 16. It is supposed to have been written by Thomas da Celano, a Franciscan friar of the thirteenth century. It was translated by Crashaw and Dryden in the seventeenth century, and by Macaulay and others in the nineteenth, but none of these translations conveys the solemn force of the original.

DIEST (d[=e]st), a town, Belgium, province of Brabant, 32 miles E.N.E. of Brussels. It has some manufactures, but the chief products of the place are beer and gin, the former being largely exported. The town was occupied by the Germans in 1914, and re-entered by the Belgians in 1918. Pop. 8800.

DIESTERWEG, Friedrich Adolf Wilhelm, German educator, born in 1790, died in 1866. In 1820 he became director of the new Teachers' Seminary at M[:o]rs, and soon gained a reputation as teacher and educator. He was a follower of Pestalozzi, and aimed at making every subject of instruction a means of education. In 1827 he founded the _Rheinische Bl[:a]tter f[:u]r Erziehung und Unterricht_, wherein he advocated his pedagogical views.

DI'ET, a meeting of some body of men held for deliberation or other purposes; a term especially applied to the legislative or administrative assemblies of Austria, Germany, and Poland.

DIETET'ICS (Gr. _diaita_, daily regimen), that part of medicine which relates to the regulation of diet. The ideal diet is clearly that which, without burdening the viscera uselessly, furnishes all necessary nutritive elements, with due consideration for special physiological conditions in any given case. Under the head of _Aliment_ the physiological properties of various foods have already been considered theoretically in respect of their capacity to supply physical waste in nitrogenous and non-nitrogenous matter. (See _Aliment_.) No single substance contains the elements needed to replace this waste in their requisite proportions, and a mixed diet is therefore necessary. For instance, to secure the required amount of carbon a man would need to eat about 4 lb. of lean beef, while 1 lb. would yield all the nitrogen required; thus, apart from the labour of digesting 4 lb. of beef, the body would be compelled to get rid of the excess of nitrogen. Bread, on the other hand, has carbon in abundance, but is deficient in nitrogen; so that by uniting 2 lb. of bread with 3/4 lb. of lean meat, the due proportion of carbon and nitrogen is satisfactorily supplied. Milk and oatmeal taken together also contain nitrogenous and non-nitrogenous substances in nearly the required proportions. A certain proportion of saline matter is also necessary. The nature of the food most suitable for a healthy man is dependent in part upon general conditions, such as climate and season, and in part upon special conditions of individual habit. The inhabitants of the Arctic regions need large quantities of oleaginous food; those of the Tropics live chiefly on starchy products. With increased

## activity and exertion, as in training, an increase in the nitrogenous foods

becomes necessary. In a state of health we need not draw hairbreadth distinctions as to the superior salubrity of the several sorts of diet, the quantity rather than the quality of food being the main consideration. Those persons who have been most remarkable for health and long life have generally been contented with two moderate meals a day, which are certainly quite sufficient during a state of health. In various countries the breakfast generally consists of tea, coffee, or cocoa, with a certain proportion of bread and butter; persons with delicate digestive powers, or who lead a sedentary life, cannot with safety or comfort eat animal food _constantly_ to breakfast. At dinner all made-dishes highly spiced, such as curries, turtle-soup, &c., as provoking appetite, are hurtful; and the custom of late dining is not to be commended. Stewed and boiled meats are more difficult to digest than meat cooked by fire alone. The flesh of young animals seems to be more difficult of digestion than that of old; and the flesh of tame than that of wild animals. All sorts of fat meat must be taken in smaller quantities. Hence, also, ham, bacon, and salted meats cannot be eaten in such quantities as the tender flesh of poultry. Fish has the advantage of being easily soluble. All boiled vegetables are in general easy of digestion; raw vegetables and salads are rather more difficult. Fruit should be taken in the forenoon rather than after a hearty meal. The moderate use of fermented liquors is far from being invariably an evil, but the smaller the quantity habitually used the better in the majority of cases.

In all diseases attended with much fever or quickness of pulse the stomach loathes animal food, and there is generally a great increase of thirst, to quench which water, either quite cold, or iced, or tepid, or rendered acid, may be freely indulged. Infusions, too, of barley, sage, balm, &c., may be taken. In chronic diseases attended with hectic fever, milk is the most proper diet. The best food for infants is, of course, their mother's milk; but whenever they begin to cut teeth a little animal food, such as soft-boiled eggs, beef-tea, and even chicken minced very fine, may be given. Many infants suffer from having too much sugar given them in their food.--BIBLIOGRAPHY: R. H. Chittenden, _Physiological Economy in Nutrition_, and _Nutrition of Man_; Hutchison, _Food and Dietetics_; Lusk, _The Science of Nutrition_.

DIETRICH (d[=e]'tri_h_), Christian Wilhelm Ernst, a German painter and engraver, born in 1712, died in 1774. He studied under his father, and afterwards under Alexander Thiele at Dresden, where he became court-painter and professor in the academy. He adopted several different manners, successfully imitating Raphael and Mieris, Correggio, and Ostade.

DIETRICH OF BERN (d[=e]'tri_h_), the name under which Theodoric the Great, King of the Ostrogoths, appears in the old German legends. Bern stands for Verona, his capital.

DIEU, or D'YEU (dy_eu_; ancient INSULA DEI), an island off the west coast of France, department of Vend['e]e. It is inaccessible on the west side, but on the east has a tolerable harbour defended by batteries. The chief industry is fishing. There are four lighthouses on the island. Pop. 3809.

DIEU ET MON DROIT (dy_eu_ e mo[n.] drw[:a]; 'God and my right'), the battle-cry of Richard I at the battle of Gisors (1198), signifying that he was not subject to France, but owed his power to God alone. The battle-cry was then adopted as the motto of the arms of England, and revived by Edward III in 1340, when he claimed the crown of France. Except during the reigns of Elizabeth and Anne, who used the motto _Semper eadem_, and of William III, who personally used _Je maintiendray_, it has ever since been the royal motto of England.

DIEZ (d[=e]ts), Friedrich Christian, German philologist of the Romance languages, born in 1794, died in 1876. Having qualified himself as a lecturer at Bonn, he was appointed professor of the Romance languages there in 1830. His work stands in much the same relation to the Romance dialects which the researches of Grimm occupy with respect to German dialects. In addition to various works on the poetry of the Troubadours, he published a very valuable _Grammatik der Romanischen Sprachen_ (1836-42, translated into English by Cayley in 1863), and an _Etymologisches W[:o]rterbuch der Romanischen Sprachen_ (1853).

DIFFERENCE, a stock-exchange term. When stock is bought or sold merely as a speculation for the rise or fall, with no intention of the buyer to 'take up' the stock, or of the seller to deliver it, the 'difference' is the movement in price which may take place between the date of the transaction and the following 'settling-day'. If the price falls, the buyer has to pay the difference upon 'carrying over' his purchase to the next account; if it rises, the seller is at the loss. Since the first weeks of the European War all stock-exchange transactions have been made, in theory at least, for cash, and speculative business of this nature has been consequently much reduced.

DIFFERENCES, FINITE, a calculus much used in actuarial work, which deals with a series of numbers by considering the differences of the successive terms.

If u_1, u_2, u_3, ... are the terms of the series, then u_2 - u_1, u_3 - u_2, u_4 - u_3,... form another series called the series of first differences. The notation used is u_2 - u_1 = [Delta]u_1, u_3 - u_2 = [Delta]u_2,...

These first differences may themselves be differenced, giving the second differences [Delta]u_2 - [Delta]u_1, [Delta]u_3 - [Delta]u_2, ..., which are written [Delta]^2u_1, [Delta]^2u_2,...

Similarly, we form the third differences [Delta]^3u_1 = [Delta]^2u_2 - [Delta]^2u_1, and so on.

As an example, let the original series be the cubes of the natural numbers.

1 8 27 64 125 216 / 343 512 7 19 37 61 91 / 127 169 12 18 24 30 / 36 42 6 6 6 / 6 6 0 0 / 0 0

Here we begin by writing down the series of cubes as far, say, as 216; beneath these we write the first differences 8 - 1 = 7, 27 - 8 = 19, &c. We thus obtain the part of the table to the left of the diagonal line.

We observe that the third differences are constant, each being 6. (It is easy to prove generally that the nth differences of the series, 1^n, 2^n, 3^n,..., are constant.) Knowing the third differences, we can now extend the table as far as we wish to the right of the diagonal line. We get first 6 + 30 = 36, 36 + 91 = 127, 127 + 216 = 343. We infer that 7^3 = 343.

Since u_1 - u_0 = [Delta]u_0, we have u_1 = u_0 + [Delta]u_0 = (1 + [Delta])u_0.

Similarly, u_2 = (1 + [Delta])u_1 = (1 + [Delta])^2u_0; and, generally,

u^x = (1 + [Delta])^xu_0 = {1 + x[Delta] + (x(x-1)/(1x2)) x [Delta]^2 + ...}u_0 = u_0 + x[Delta]u_0 + (x(x-1)/(1x2)) x [Delta]^2)u_0 + ...

a formula much used by calculators, and known as Newton's interpolation formula.

The above symbolic method of proof only applies when x is a positive integer, but the result is used in practice even for fractional values of x, as in most cases the high differences become negligible.

If n is a positive integer, it is easy to prove that

[Delta]^nu_x = u_{x+n} - nu_{x+n-1} + ((n(n-1))/(1x2))u_{x+n-2} - ...

If the nth differences vanish, or are negligible, this gives

0 = u_{x+n} - nu_{x+n-1} + ((n(n-1))/(1x2))u_{x+n-2} -... +(-1)^nu_x,

another useful interpolation formula, by which we can calculate any missing term of a series.--BIBLIOGRAPHY: G. Boole, _Finite Differences; Textbook of the Institute of Actuaries_.

DIFFERENTIAL EQUATION, an algebraical relation involving derivatives or differentials. Examples:

d^2z/dt^2 = g: ydx + xdy + zdz = 0.

An _ordinary_ differential equation involves only one independent variable, a _partial_ differential equation involves more than one. Examples of ordinary equations:

d^2y/dx^2 + (1/x)dy/dx + y = 0; d^2x/dt^2 + a(dy/dt) + px + qy = 0.

Examples of partial differential equations:

x(dz/dx) + y(dz/dy) = nz; d^2u/dx^2 + d^2u/dy^2 + d^2u/dz^2 = 0.

Equations, whether ordinary or partial, can also be classified as _linear_ or _non-linear_. A linear equation is a rational integral equation of the first degree in the dependent variable or variables and their derivatives. The equation

x^2 d^2y/dx^2 + x dy/dx + (x^4 + 1)y = 0

is linear, but

(dy/dx)^2 = xy and y dy/dx = x^2

are non-linear. The _order_ of an equation is the order of the highest derivative or differential which it contains. Of the three equations last written, the first is _linear of the second order_, the other two are of the _first order and second degree_. To _integrate_ a differential equation or system of equations is to find a relation or relations among the variables, equivalent to the given equation or equations. Thus the integral of

d^2z/dt^2 = g is z = 1/2gt^2 + At + B,

where A and B are _arbitrary constants_. An ordinary equation of the _n_th order with one dependent variable has exactly _n_ arbitrary constants in its complete integral, or solution. In a practical problem the arbitrary constants are determined by the _initial_, or _boundary_, _conditions_. The solution of d^2z/dt^2 = g, e.g. is completely determinate if the values of z and dz/dt when t = 0 are given. The solution of _partial_ equations may involve arbitrary _functions_, which become definite when proper initial or boundary conditions are assigned. Thus the equation du/dx = du/dt has for its complete solution u = [phi](x + t), where [phi] may be a function of any form whatever; if now we are given that, when t = 0, u = a given function f(x), we obtain f(x) = [phi](x), so that the solution required is u = f(x + t). Certain ordinary linear equations of the second order are specially important, both from the beauty of their theory and from their usefulness in Mathematical Physics. Some of these equations are: Bessel's equation, Legendre's equation, the hypergeometric equation, Mathieu's equation, Lam['e]'s equation. Linear partial equations of the second order are fundamental in Physics. Such are: Laplace's equation,

d^2V/dx^2 + d^2V/dy^2 + d^2V/dz^2= 0;

the wave equation,

d^2V/dt^2 = c^2(d^2V/dx^2 + d^2V/dy^2 + d^2V/dz^2);

the equation of conduction of heat,

dV/dt = k(d^2V/dx^2 + d^2V/dy^2 + d^2V/dz^2).

These involve one dependent variable only. Equations with several dependent variables occur in Elasticity, Electrodynamics, and Hydrodynamics. A notable feature of the hydrodynamical equations is that they are not linear.

No general rules exist enabling us to deal with a differential equation taken at random, and only a few types have been completely solved. Of soluble equations, the most important are those which are _linear with constant coefficients_.

_Example 1._ d^2x/dt^2 - 7dx/dt + 12 = 0. To solve this, try x = e^{mt}. We find e^{mt}(m^2 - 7m + 12) = 0. Thus m = 3 or 4. It is now easy to show that x = Ae^{3t} + Be^{4t} is a solution, where A and B are arbitrary constants. This is the general solution. We can determine A and B if the values of x and dx/dt are given for a definite value of t, say t = 0.

_Example 2._ d^2y/dt^2 = c^2 d^2y/dx^2. Try y = e^{lx + mt}. We find m^2 = c^2l^2, or m = +/-cl. Hence y = Ae^{l(x + ct)} + Be^{l(x - ct)} is a solution for all values of A, B, l; so, also, is the sum of any number of terms of similar forms. We may infer that the general solution is

y = f(x + ct) + F(x - ct),

where f and F are arbitrary functions. It is only in exceptional cases that an equation can be solved, as in these two examples, by an analytical formula; indeed, differential equations are the most fertile source of new functions in analysis. But, as in the analogous cases of algebraic equations and definite integrals, it may be quite possible to find, by methods of approximation, an arithmetical solution which is sufficient for the purpose in hand.--BIBLIOGRAPHY: H. T. H. Piaggio, _Differential Equations_; J. M. Page, _Ordinary Differential Equations_; A. R. Forsyth, _A Treatise on Differential Equations_; E. T. Whittaker and G. N. Watson, _Modern Analysis_.

DIFFRAC'TION, a term applied to the bending that rays of light undergo in passing close to the edge of an opaque body. Thus when a beam of direct sunlight is admitted into a dark room through a narrow slit, and falls upon a screen placed to receive it, there appears a line of white light bordered by coloured fringes; these fringes are produced by diffraction, and in the case given it may be seen that the red or long-wave rays are diffracted more than the blue rays. See _Interference_.

DIFFU'SION, the gradual mixing of gases, liquids, or solids when brought into direct contact. When a block of lead is placed on a block of gold, with their smooth surfaces in close contact, it is found that, after several weeks, gold has diffused into the lead, and lead into the gold. In the case of gases, when a jar of oxygen and a jar of hydrogen are connected together by a tube or opening of any kind, they rapidly become mixed; and their mixture does not depend on gravity, but takes place in opposition to that force, as may be shown by placing the jar of hydrogen gas above the other. Oxygen is sixteen times heavier than hydrogen, bulk for bulk, but the heavier gas moves upwards and the lighter downwards, and the process of intermixture, or _diffusion_, goes on till the two gases are apparently equably distributed throughout the whole space. After that they have no tendency whatever to separate. Similarly, if two vessels, one containing oxygen and the other hydrogen, be connected by a tube which is stuffed with a plug of porous material, such as plaster of Paris, the gases gradually diffuse one into the other through the porous plug. The two gases, however, do not pass through the porous separator at equal rates, but in _inverse proportion to the square roots of the densities of the gases_. Thus in the case of two vessels, one containing hydrogen and the other oxygen, which is sixteen times as heavy as hydrogen, the hydrogen will pass towards the oxygen jar four times as quickly as the oxygen will pass towards the hydrogen jar. Kindred phenomena occur when two liquids that are capable of mixing, such as alcohol and water, are put in contact, the two gradually diffusing one into the other in spite of the action of gravity. In some cases, however, as where ether and water are employed, the diffusion is only partial, this result arising from the fact that these two liquids are not miscible in all proportions. When solutions of various solid bodies are placed in contact, interdiffusion also takes place. On the results of his examination of the phenomena of diffusion of liquids and salts across porous membranes or _septa_, Graham founded a method of separating _colloid_ from _crystalloid_ bodies, which he called _dialysis_.

DIGAM'MA, a letter which once belonged to the Greek alphabet, and which remained longest in use among the Aeolians. It resembled our letter F, and hence was called _digamma_, that is, double [Gamma]. It appears to have had the force of _f_ or _v_. Its existence was first pointed out by Richard Bentley.

DIGBY, Sir Everard, an English gentleman, born of a Roman Catholic family in 1578. He enjoyed some consideration at the court of Elizabeth and James I, by whom he was knighted. Having contributed money to the Guy Fawkes conspiracy, he was tried and hanged in 1606.

DIGBY, Sir Kenelm, eldest son of the preceding, born in 1603, died in 1665. He studied at Oxford, was knighted in 1623, and on the accession of Charles I was created a gentleman of the bedchamber, a Commissioner of the Navy, and a governor of Trinity House. He soon after fitted out at his own expense a small but successful squadron against the French and Venetians. In 1636 he became a Roman Catholic, and was imprisoned as a Royalist during 1642-3, when he was allowed to retire to the Continent. At the Restoration he returned to England, became a member of the Royal Society, and was much visited by men of science. He wrote numerous works: a _Treatise on the Nature of Bodies_, a _Treatise on the Nature and Operation of the Soul_, and _Of the Cure of Wounds by the Powder of Sympathy_.

DI'GEST, a name originally given to a collection or body of Roman laws, digested or arranged under proper titles by order of the Emperor Justinian. Hence applied to any somewhat similar collection.